## A Filter Active-Set Trust-Region Method (2007)

Citations: | 3 - 0 self |

### BibTeX

@MISC{Friedl07afilter,

author = {Michael P. Friedl and Nick I. M. Gould and Sven Leyffer and Todd S. Munson},

title = {A Filter Active-Set Trust-Region Method},

year = {2007}

}

### OpenURL

### Abstract

2.1 Sequential Linear-Quadratic Programming Methods.............. 3 2.2 Difficulties with the LP/TR Step Computation................ 4

### Citations

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Citation Context ...e return to this issue below, but assume for now that one can find a solution dLP to RLP(x, µ). The problem RLP(x, µ) has an interesting history. In particular, RLP(x, µ) is a Tikhonov regularization =-=[27]-=- of LP(x), for which the solution of the former is the least ℓ2-norm solution to the latter for all µ sufficiently large [22]. See [29] for further references. Vitally, since the Hessian of the object... |

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Citation Context ...∆f = f(x) − f(x + d) be the actual reduction in f(x) resulting from a given step d. We measure constraint infeasibility by h(c(x)) = �c(x) + �∞, where ci(x) + = max(0, −ci(x)). We now define a filter =-=[10]-=- that combines the two competing aims in NLP, namely, minimization of f(x) and h(c(x)). We will consider pairs (h, f) obtained by evaluating f(x) and h(c(x)). We say that a pair (hk, fk) dominates ano... |

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Citation Context ...atter for all µ sufficiently large [22]. See [29] for further references. Vitally, since the Hessian of the objective is diagonal, the dual of RLP(x, µ) is a bound-constrained quadratic program (BQP) =-=[16, 21]-=- in the dual variables y, (BQP(x, µ)) � minimize y subject to y ≥ 0, 1 2yT AT (x)A(x)y + � c(x) − µAT (x)g(x) �T µ y + 2 2 gT (x)g(x) from which the solution d = −µg(x)+A(x)y of RLP(x, µ) may be recov... |

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Citation Context ... appropriate, but the downside is that TR(x, ρ) is no longer an LP and may be more difficult to solve—there has been some work for the ℓ2-norm case using semi-definite and cone programming techniques =-=[1, 12, 17, 18, 20, 26]-=-, but to our knowledge the size of problems that are amenable to such techniques fall far short of those currently possible for LP subproblems. A second difficulty, common to many trust-region methods... |

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Citation Context ... � c(x) − µAT (x)g(x) �T µ y + 2 2 gT (x)g(x) from which the solution d = −µg(x)+A(x)y of RLP(x, µ) may be recovered. The significance is that there are excellent methods [7, 15, 21, 23] and software =-=[8, 19]-=- for solving large-scale BQPs. Having solved RLP(x, µ)—or its dual, BQP(x, µ)—to find an initial step dLP and its associated active set A, our algorithm solves the resulting EQP(x, A) to find a second... |

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Citation Context ...m an excellent guess of the optimal solution. Interior-point methods are not readily warm-started and are not as efficiently warm-started as are active-set methods, despite recent efforts in the area =-=[2, 11, 13]-=-. PDE-constrained optimization problems give rise to huge linear systems that must be solved at every iteration. In particular, the linear systems arising from any optimization method can no longer be... |

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Citation Context ... � c(x) − µAT (x)g(x) �T µ y + 2 2 gT (x)g(x) from which the solution d = −µg(x)+A(x)y of RLP(x, µ) may be recovered. The significance is that there are excellent methods [7, 15, 21, 23] and software =-=[8, 19]-=- for solving large-scale BQPs. Having solved RLP(x, µ)—or its dual, BQP(x, µ)—to find an initial step dLP and its associated active set A, our algorithm solves the resulting EQP(x, A) to find a second... |

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Citation Context ...to y ≥ 0, 1 2yT AT (x)A(x)y + � c(x) − µAT (x)g(x) �T µ y + 2 2 gT (x)g(x) from which the solution d = −µg(x)+A(x)y of RLP(x, µ) may be recovered. The significance is that there are excellent methods =-=[7, 15, 21, 23]-=- and software [8, 19] for solving large-scale BQPs. Having solved RLP(x, µ)—or its dual, BQP(x, µ)—to find an initial step dLP and its associated active set A, our algorithm solves the resulting EQP(x... |

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Citation Context ...interesting history. In particular, RLP(x, µ) is a Tikhonov regularization [27] of LP(x), for which the solution of the former is the least ℓ2-norm solution to the latter for all µ sufficiently large =-=[22]-=-. See [29] for further references. Vitally, since the Hessian of the objective is diagonal, the dual of RLP(x, µ) is a bound-constrained quadratic program (BQP) [16, 21] in the dual variables y, (BQP(... |

49 | On the global convergence of an SLP-filter algorithm that takes EQP steps
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Citation Context ...y the optimal active set. Next, we describe our globalization strategy and provide a formal statement of our new algorithm. 2.1 Sequential Linear-Quadratic Programming Methods We believe SLQP methods =-=[3,4,6,24]-=- are well suited for large-scale nonlinear programming, simply because each of the key steps involved—the solution an LP or EQP, together with its attendant linear algebra—has already been tuned to co... |

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Citation Context ...atter for all µ sufficiently large [22]. See [29] for further references. Vitally, since the Hessian of the objective is diagonal, the dual of RLP(x, µ) is a bound-constrained quadratic program (BQP) =-=[16, 21]-=- in the dual variables y, (BQP(x, µ)) � minimize y subject to y ≥ 0, 1 2yT AT (x)A(x)y + � c(x) − µAT (x)g(x) �T µ y + 2 2 gT (x)g(x) from which the solution d = −µg(x)+A(x)y of RLP(x, µ) may be recov... |

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Citation Context ...y the optimal active set. Next, we describe our globalization strategy and provide a formal statement of our new algorithm. 2.1 Sequential Linear-Quadratic Programming Methods We believe SLQP methods =-=[3,4,6,24]-=- are well suited for large-scale nonlinear programming, simply because each of the key steps involved—the solution an LP or EQP, together with its attendant linear algebra—has already been tuned to co... |

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Citation Context ...e active set for one subproblem is a good prediction for that of the next (and thus allows efficient “warm starts”). Extensive numerical experience with the SLIQUE and KNITRO-ACTIVE software packages =-=[3, 5]-=- has indicated that indeed in some cases warm-start strategies may be inefficient for this reason. One way to avoid this drawback is to use a trust-region norm that does not have multiple faces. Then,... |

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Citation Context ...y the optimal active set. Next, we describe our globalization strategy and provide a formal statement of our new algorithm. 2.1 Sequential Linear-Quadratic Programming Methods We believe SLQP methods =-=[3,4,6,24]-=- are well suited for large-scale nonlinear programming, simply because each of the key steps involved—the solution an LP or EQP, together with its attendant linear algebra—has already been tuned to co... |

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Citation Context ... appropriate, but the downside is that TR(x, ρ) is no longer an LP and may be more difficult to solve—there has been some work for the ℓ2-norm case using semi-definite and cone programming techniques =-=[1, 12, 17, 18, 20, 26]-=-, but to our knowledge the size of problems that are amenable to such techniques fall far short of those currently possible for LP subproblems. A second difficulty, common to many trust-region methods... |

12 |
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Citation Context ... appropriate, but the downside is that TR(x, ρ) is no longer an LP and may be more difficult to solve—there has been some work for the ℓ2-norm case using semi-definite and cone programming techniques =-=[1, 12, 17, 18, 20, 26]-=-, but to our knowledge the size of problems that are amenable to such techniques fall far short of those currently possible for LP subproblems. A second difficulty, common to many trust-region methods... |

12 | A new unblocking technique to warmstart interior point methods based on sensitivity analysis
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Citation Context ...m an excellent guess of the optimal solution. Interior-point methods are not readily warm-started and are not as efficiently warm-started as are active-set methods, despite recent efforts in the area =-=[2, 11, 13]-=-. PDE-constrained optimization problems give rise to huge linear systems that must be solved at every iteration. In particular, the linear systems arising from any optimization method can no longer be... |

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Citation Context ...g history. In particular, RLP(x, µ) is a Tikhonov regularization [27] of LP(x), for which the solution of the former is the least ℓ2-norm solution to the latter for all µ sufficiently large [22]. See =-=[29]-=- for further references. Vitally, since the Hessian of the objective is diagonal, the dual of RLP(x, µ) is a bound-constrained quadratic program (BQP) [16, 21] in the dual variables y, (BQP(x, µ)) � m... |

4 |
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Citation Context ...to y ≥ 0, 1 2yT AT (x)A(x)y + � c(x) − µAT (x)g(x) �T µ y + 2 2 gT (x)g(x) from which the solution d = −µg(x)+A(x)y of RLP(x, µ) may be recovered. The significance is that there are excellent methods =-=[7, 15, 21, 23]-=- and software [8, 19] for solving large-scale BQPs. Having solved RLP(x, µ)—or its dual, BQP(x, µ)—to find an initial step dLP and its associated active set A, our algorithm solves the resulting EQP(x... |

4 |
On the global convergence of an SLP-filter algorithm. Numerical Analysis Report NA/183
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Citation Context ...d Figure 4: FASTr: Filter Active-Set Trust-Region Methods12 Michael Friedlander, Nick Gould, Sven Leyffer, and Todd Munson 3 Convergence Analysis The global convergence proof is similar to the one in =-=[25]-=- and shows that the iterates generated by FASTr converge to stationary point. In fact, taking RLP steps is sufficient for global convergence. However, we also show that the convergence of the EQP step... |

2 |
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Citation Context ...m an excellent guess of the optimal solution. Interior-point methods are not readily warm-started and are not as efficiently warm-started as are active-set methods, despite recent efforts in the area =-=[2, 11, 13]-=-. PDE-constrained optimization problems give rise to huge linear systems that must be solved at every iteration. In particular, the linear systems arising from any optimization method can no longer be... |

1 |
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Citation Context ...step dSOC are intended to accelerate convergence and thus to be preferred if suitable. We also note that the computation of superficially similar steps has recently been proposed by Yamashita and Dan =-=[28]-=- but with significant differences, the most important being that µ is not used as a control parameter as it is here and that a different globalization strategy is used. 2.4 Globalization and the Filte... |